- The Hamiltonian for several nuclei and electrons would be
A runs over nuclei, i over electrons. Note that unlike an atom in free space,
we now have a term in nuclear motion. Wave function would be
- Nuclei are much heavier than electrons (at least 2000 times).
Decouple problem: assert that there is no coupling between electornic
motion (fast) and nuclear motion (slow).
- Write wave function as
where we have a different electronic wave function for each value of the
parameters RA - coordinates of fixed nuclei. Eigenfunction of
- Electronic Schrödinger equation
the function E(RA) defines the potential energy surface for nuclear
- Get rotational and vibrational energy levels for the nuclear part of the
problem, by solving
the nuclear motion wave function. Also determines chemical dynamics.
- Note that this is an approximation - properly speaking there are just
eigenvalues of the full Hamiltonian, representing "rovibronic" levels. But a
very good one.
- Fixing the nuclei has a drastic effect on the symmetry of the system!
- Even with the Born-Oppenheimer approimxation, we can make no progress on
analytical solution of the Schrödinger equation.
- Start simple: assume traditional molecular orbital ideas hold. For a system
of 2N electrons, assume tha the wave function is given by allocating electrons
in pairs to molecular orbitals
- Physically, as we shall see, this is like assuming that the electrons interact
only with the average potential of all the other electrons.
Independent-particle model, or mean-field theory.
- Assume that the MOs are orthonormal (and here real)
- Trial wave function is a Slater determinant
Substitute into the variation principle, and make stationary with respect to
the form of the MOs.
- Need to evaluate
for this form of
Helped here by the orthonormality of the MOs and by the form of H.
- H contains zero-, one-, and two-electron operators only. In the integral over
all space, we have contributions from the coordinates of N-2 electrons that are
just products of integrals
so thanks to orthnormality we end up with only one- and two-electron
- The expression
is the energy for a given guess at the MOs. We optimize the MOs by minimizing
this energy (variational method).
- Minimizing the energy requires satisfying the Hartree-Fock equations
is the Fock operator.
- Notes the sum over all k here. The solutions depend on themselves! This is
not a simple eigenvalue problem.
- Self-consistent field (SCF) approach: guess a sert of MOs, construct
F, diagonalize, and use its eigenvectors as a (hopefully) better guess
at the MOs.
Solving the SCF equations
- For atoms, the symmetry of the system factors the SCF equations into an angular
part (spherical harmonics again!) and a radial part. The radial equations can
be solved numerically (one-dimensional problem).
- For molecules, the lower symmetry does not permit any factorization in general.
Numerical methods have been used for diatomic molecules, but becomes expensive
and cumbersome for polyatomic systems.
- How do we then express the MOs? Expand them in a fixed basis set, just like
the traditional LCAO (linear combination of atomic orbitals) reasoning used in
qualitative MO theory.
The SCF equations
- Choose a basis
- Calculate the integrals over this basis (store on disk).
- Guess the MO coefficients C.
- Construct D, then F. Solve the equations for a new C. If
it does not agree with the previous estimate, iterate until it does.